M2: Analysis II - Continuity and Differentiability (2026-27)
Main content blocks
- Lecturer: Profile: Richard Earl
Sequences and series of functions, uniform convergence. Weierstrass's M-test for uniformly convergent series of functions. Uniform limit of a sequence of continuous functions is continuous. Continuity of functions defined by power series.
Definition of the derivative of a function of a real variable. Algebra of derivatives, examples to include polynomials and inverse functions. The derivative of a function defined by a power series is given by the derived series (proof not examinable). Vanishing of the derivative at a local maximum or minimum. Rolle's Theorem, Mean Value Theorem, and Cauchy's (Generalized) Mean Value Theorem with applications: Constancy Theorem, monotone functions, exponential function and trigonometric functions. L'Hôpital's Formula. Taylor's Theorem with remainder in Lagrange's form; examples. The binomial expansion with arbitrary index.
Section outline
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We discuss limits and continuity of functions - initially real functions on real intervals, and then generalizing to other domains.
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We meet several important theorems from the 19th century about continuous real functions on intervals.
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We discuss pointwise and uniform convergence of functions, showing further that power series define continuous functions on their disc of convergence.
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We define differentiability, prove Leibniz's rules on the algebra of differentiation, and some early results - Fermat, Rolle, MVT, DIFT.
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Continuity and Limits on Intervals. This covers material from the first two lectures.
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We extend the notion of continuity to general domains in R and C. Then introduce limit points so that limits can be defined on general domains too.
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We meet some first theorems about continuous functions on a closed bounded interval: the IVT and the boundedness theorem.
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Problems concerning the invertibility of continuous functions and uniform continuity.
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Problems largely on pointwise and uniform convergence, with some introductory exercises on differentiability.
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